a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__U52(tt, V2) → A__U53(a__isNatList(V2))
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U43(X)) → A__U43(mark(X))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U11(tt, V1) → A__U12(a__isNatList(V1))
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
A__U61(tt, L) → MARK(L)
MARK(zeros) → A__ZEROS
MARK(s(X)) → MARK(X)
MARK(U12(X)) → A__U12(mark(X))
A__U21(tt, V1) → A__U22(a__isNat(V1))
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__U42(tt, V2) → A__U43(a__isNatIList(V2))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
MARK(U22(X)) → A__U22(mark(X))
A__U31(tt, V) → A__U32(a__isNatList(V))
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U53(X)) → A__U53(mark(X))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(cons(X1, X2)) → MARK(X1)
A__U52(tt, V2) → A__U53(a__isNatList(V2))
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U43(X)) → A__U43(mark(X))
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U11(tt, V1) → A__U12(a__isNatList(V1))
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
A__U61(tt, L) → MARK(L)
MARK(zeros) → A__ZEROS
MARK(s(X)) → MARK(X)
MARK(U12(X)) → A__U12(mark(X))
A__U21(tt, V1) → A__U22(a__isNat(V1))
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__U42(tt, V2) → A__U43(a__isNatIList(V2))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
MARK(U22(X)) → A__U22(mark(X))
A__U31(tt, V) → A__U32(a__isNatList(V))
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U53(X)) → A__U53(mark(X))
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U53(X)) → MARK(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
MARK(U22(X)) → MARK(X)
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
A__U61(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__ISNATILIST(V) → A__ISNATILISTKIND(V)
A__ISNATILIST(V) → A__U31(a__isNatIListKind(V), V)
A__ISNATILIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → MARK(X1)
A__U41(tt, V1, V2) → A__ISNAT(V1)
MARK(U42(X1, X2)) → MARK(X1)
A__ISNATILIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
Used ordering: Polynomial interpretation [25,35]:
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U53(X)) → MARK(X)
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
A__U61(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)
The value of delta used in the strict ordering is 4.
POL(A__ISNATILISTKIND(x1)) = 0
POL(A__U31(x1, x2)) = 0
POL(A__ISNAT(x1)) = 0
POL(a__U11(x1, x2)) = (4)x_1
POL(a__zeros) = 0
POL(U41(x1, x2, x3)) = 1 + (4)x_1
POL(mark(x1)) = x_1
POL(a__isNatIList(x1)) = 1
POL(A__ISNATLIST(x1)) = 0
POL(A__AND(x1, x2)) = (4)x_2
POL(a__length(x1)) = (4)x_1
POL(tt) = 0
POL(A__U42(x1, x2)) = 4 + x_1
POL(a__U21(x1, x2)) = x_1
POL(nil) = 0
POL(U31(x1, x2)) = (4)x_1
POL(A__U11(x1, x2)) = x_1
POL(isNatKind(x1)) = 0
POL(a__U32(x1)) = (2)x_1
POL(U22(x1)) = x_1
POL(a__U31(x1, x2)) = (4)x_1
POL(A__ISNATILIST(x1)) = 4
POL(a__U52(x1, x2)) = x_1
POL(U53(x1)) = (4)x_1
POL(U32(x1)) = (2)x_1
POL(length(x1)) = (4)x_1
POL(a__U22(x1)) = x_1
POL(U52(x1, x2)) = x_1
POL(a__U53(x1)) = (4)x_1
POL(a__U42(x1, x2)) = 1 + x_1
POL(A__ISNATKIND(x1)) = 0
POL(A__U51(x1, x2, x3)) = 0
POL(and(x1, x2)) = (4)x_1 + (4)x_2
POL(A__U61(x1, x2)) = (4)x_2
POL(isNatIListKind(x1)) = 0
POL(U21(x1, x2)) = x_1
POL(isNatList(x1)) = 0
POL(zeros) = 0
POL(isNatIList(x1)) = 1
POL(U12(x1)) = x_1
POL(s(x1)) = x_1
POL(a__isNat(x1)) = 0
POL(A__U52(x1, x2)) = 0
POL(a__U43(x1)) = x_1
POL(isNat(x1)) = 0
POL(a__isNatList(x1)) = 0
POL(U51(x1, x2, x3)) = (4)x_1
POL(A__LENGTH(x1)) = (2)x_1
POL(a__U61(x1, x2)) = (4)x_1 + (4)x_2
POL(U42(x1, x2)) = 1 + x_1
POL(a__and(x1, x2)) = (4)x_1 + (4)x_2
POL(U11(x1, x2)) = (4)x_1
POL(U43(x1)) = x_1
POL(0) = 0
POL(a__U12(x1)) = x_1
POL(a__U51(x1, x2, x3)) = (4)x_1
POL(cons(x1, x2)) = x_1 + (2)x_2
POL(a__U41(x1, x2, x3)) = 1 + (4)x_1
POL(A__U21(x1, x2)) = 0
POL(MARK(x1)) = (4)x_1
POL(a__isNatKind(x1)) = 0
POL(U61(x1, x2)) = (4)x_1 + (4)x_2
POL(a__isNatIListKind(x1)) = 0
POL(A__U41(x1, x2, x3)) = 4 + (4)x_1
a__U21(X1, X2) → U21(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__U22(X) → U22(X)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U43(tt) → tt
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U22(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U32(tt) → tt
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__U61(tt, L) → s(a__length(mark(L)))
a__U53(tt) → tt
a__isNat(0) → tt
a__isNatList(nil) → tt
a__isNatKind(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U32(X)) → a__U32(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U22(X)) → a__U22(mark(X))
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(length(X)) → a__length(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIList(X)) → A__ISNATILIST(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U41(X1, X2, X3)) → A__U41(mark(X1), X2, X3)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U42(X1, X2)) → A__U42(mark(X1), X2)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
A__U61(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
A__U42(tt, V2) → A__ISNATILIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
A__U42(tt, V2) → A__ISNATILIST(V2)
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__ISNATILIST(cons(V1, V2)) → A__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__U41(tt, V1, V2) → A__U42(a__isNat(V1), V2)
Used ordering: Polynomial interpretation [25,35]:
A__U42(tt, V2) → A__ISNATILIST(V2)
The value of delta used in the strict ordering is 1.
POL(U41(x1, x2, x3)) = 3 + (4)x_1 + (2)x_2 + (3)x_3
POL(a__zeros) = 3
POL(a__U11(x1, x2)) = 3 + (3)x_1 + (2)x_2
POL(mark(x1)) = 3
POL(a__isNatIList(x1)) = 1 + (3)x_1
POL(a__length(x1)) = 3 + (2)x_1
POL(tt) = 2
POL(A__U42(x1, x2)) = (4)x_2
POL(a__U21(x1, x2)) = 3 + (3)x_1 + (4)x_2
POL(nil) = 3
POL(U31(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(isNatKind(x1)) = 3 + (3)x_1
POL(a__U32(x1)) = 4 + (3)x_1
POL(U22(x1)) = 3 + (2)x_1
POL(a__U52(x1, x2)) = 2 + (3)x_1 + (4)x_2
POL(A__ISNATILIST(x1)) = (4)x_1
POL(a__U31(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(U53(x1)) = 3 + (2)x_1
POL(U32(x1)) = 2 + x_1
POL(length(x1)) = 1 + (3)x_1
POL(a__U22(x1)) = 3
POL(U52(x1, x2)) = 4 + (2)x_1 + (3)x_2
POL(a__U53(x1)) = 3 + x_1
POL(a__U42(x1, x2)) = 2 + (3)x_1 + (3)x_2
POL(and(x1, x2)) = 3 + (2)x_1 + (3)x_2
POL(isNatIListKind(x1)) = 4 + (3)x_1
POL(U21(x1, x2)) = 3 + (2)x_1 + (4)x_2
POL(isNatList(x1)) = 1 + (3)x_1
POL(zeros) = 0
POL(isNatIList(x1)) = 1 + (2)x_1
POL(U12(x1)) = 3 + (3)x_1
POL(a__isNat(x1)) = 0
POL(s(x1)) = 1 + (2)x_1
POL(isNat(x1)) = 3 + x_1
POL(a__U43(x1)) = 3
POL(a__isNatList(x1)) = 3 + (3)x_1
POL(U51(x1, x2, x3)) = 3 + (4)x_1 + (3)x_2 + (3)x_3
POL(a__U61(x1, x2)) = 4 + (3)x_1 + (3)x_2
POL(U42(x1, x2)) = 2 + (3)x_1 + (2)x_2
POL(a__and(x1, x2)) = 2
POL(U11(x1, x2)) = 2 + (3)x_1 + (2)x_2
POL(0) = 1
POL(a__U12(x1)) = 3 + x_1
POL(U43(x1)) = 3
POL(a__U51(x1, x2, x3)) = 2 + (3)x_1 + (3)x_2 + x_3
POL(a__U41(x1, x2, x3)) = 4 + (2)x_1 + (3)x_2 + x_3
POL(cons(x1, x2)) = 3 + x_1 + (4)x_2
POL(a__isNatKind(x1)) = 4
POL(U61(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(a__isNatIListKind(x1)) = 3 + (2)x_1
POL(A__U41(x1, x2, x3)) = 1 + (4)x_3
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
A__U42(tt, V2) → A__ISNATILIST(V2)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(length(X)) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U53(X)) → MARK(X)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U32(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
A__U61(tt, L) → MARK(L)
MARK(s(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__LENGTH(cons(N, L)) → A__AND(a__isNatList(L), isNatIListKind(L))
MARK(length(X)) → MARK(X)
A__LENGTH(cons(N, L)) → A__AND(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
A__U61(tt, L) → MARK(L)
MARK(length(X)) → A__LENGTH(mark(X))
MARK(U61(X1, X2)) → A__U61(mark(X1), X2)
A__LENGTH(cons(N, L)) → A__ISNATLIST(L)
MARK(U61(X1, X2)) → MARK(X1)
Used ordering: Polynomial interpretation [25,35]:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
A__U61(tt, L) → A__LENGTH(mark(L))
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__AND(tt, X) → MARK(X)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U53(X)) → MARK(X)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
MARK(U32(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(U21(X1, X2)) → MARK(X1)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U43(X)) → MARK(X)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__U21(tt, V1) → A__ISNAT(V1)
The value of delta used in the strict ordering is 1.
POL(A__ISNATILISTKIND(x1)) = 0
POL(A__U31(x1, x2)) = (4)x_1
POL(A__ISNAT(x1)) = 0
POL(a__U11(x1, x2)) = (2)x_1
POL(U41(x1, x2, x3)) = x_2 + (4)x_3
POL(a__zeros) = 0
POL(mark(x1)) = (4)x_1
POL(a__isNatIList(x1)) = x_1
POL(A__ISNATLIST(x1)) = 0
POL(A__AND(x1, x2)) = (4)x_2
POL(a__length(x1)) = 4 + x_1
POL(tt) = 0
POL(a__U21(x1, x2)) = (4)x_1
POL(nil) = 3
POL(U31(x1, x2)) = (4)x_1
POL(A__U11(x1, x2)) = 0
POL(isNatKind(x1)) = 0
POL(a__U32(x1)) = (2)x_1
POL(U22(x1)) = (2)x_1
POL(a__U31(x1, x2)) = (4)x_1
POL(a__U52(x1, x2)) = (2)x_1
POL(U53(x1)) = (4)x_1
POL(U32(x1)) = (2)x_1
POL(length(x1)) = 4 + x_1
POL(a__U22(x1)) = (2)x_1
POL(U52(x1, x2)) = (2)x_1
POL(a__U53(x1)) = (4)x_1
POL(A__ISNATKIND(x1)) = 0
POL(a__U42(x1, x2)) = (3)x_2
POL(A__U51(x1, x2, x3)) = 0
POL(and(x1, x2)) = (2)x_1 + x_2
POL(isNatIListKind(x1)) = 0
POL(A__U61(x1, x2)) = 1 + (4)x_2
POL(U21(x1, x2)) = (4)x_1
POL(isNatList(x1)) = 0
POL(zeros) = 0
POL(isNatIList(x1)) = x_1
POL(U12(x1)) = (2)x_1
POL(s(x1)) = x_1
POL(a__isNat(x1)) = 0
POL(A__U52(x1, x2)) = (2)x_1
POL(a__U43(x1)) = (2)x_1
POL(isNat(x1)) = 0
POL(a__isNatList(x1)) = 0
POL(U51(x1, x2, x3)) = x_1
POL(A__LENGTH(x1)) = 1 + x_1
POL(a__U61(x1, x2)) = 4 + (2)x_1 + (4)x_2
POL(a__and(x1, x2)) = (2)x_1 + (4)x_2
POL(U42(x1, x2)) = (3)x_2
POL(U11(x1, x2)) = (2)x_1
POL(0) = 0
POL(U43(x1)) = (2)x_1
POL(a__U12(x1)) = (2)x_1
POL(a__U51(x1, x2, x3)) = x_1
POL(a__U41(x1, x2, x3)) = x_2 + (4)x_3
POL(cons(x1, x2)) = x_1 + (4)x_2
POL(A__U21(x1, x2)) = x_1
POL(MARK(x1)) = (4)x_1
POL(a__isNatKind(x1)) = 0
POL(U61(x1, x2)) = 2 + (2)x_1 + x_2
POL(a__isNatIListKind(x1)) = 0
a__U21(X1, X2) → U21(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__U22(X) → U22(X)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U43(tt) → tt
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U22(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U32(tt) → tt
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__U61(tt, L) → s(a__length(mark(L)))
a__U53(tt) → tt
a__isNat(0) → tt
a__isNatList(nil) → tt
a__isNatKind(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U32(X)) → a__U32(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U22(X)) → a__U22(mark(X))
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(length(X)) → a__length(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
MARK(U12(X)) → MARK(X)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
A__U61(tt, L) → A__LENGTH(mark(L))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__AND(tt, X) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(isNat(X)) → A__ISNAT(X)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__U21(tt, V1) → A__ISNAT(V1)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
A__U61(tt, L) → A__LENGTH(mark(L))
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__U61(tt, L) → A__LENGTH(mark(L))
Used ordering: Polynomial interpretation [25,35]:
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
The value of delta used in the strict ordering is 2.
POL(U41(x1, x2, x3)) = (4)x_3
POL(a__zeros) = 1
POL(a__U11(x1, x2)) = x_1
POL(mark(x1)) = 1 + (2)x_1
POL(a__isNatIList(x1)) = 3 + (4)x_1
POL(a__length(x1)) = 0
POL(tt) = 1
POL(a__U21(x1, x2)) = 1
POL(nil) = 1
POL(U31(x1, x2)) = 1
POL(isNatKind(x1)) = 0
POL(a__U32(x1)) = 1
POL(U22(x1)) = 0
POL(a__U52(x1, x2)) = (4)x_2
POL(a__U31(x1, x2)) = 3
POL(U53(x1)) = x_1
POL(U32(x1)) = 1
POL(length(x1)) = 0
POL(a__U22(x1)) = 1
POL(U52(x1, x2)) = (2)x_2
POL(a__U53(x1)) = x_1
POL(a__U42(x1, x2)) = x_1 + x_2
POL(and(x1, x2)) = x_1 + (2)x_2
POL(isNatIListKind(x1)) = 0
POL(A__U61(x1, x2)) = (4)x_1 + (4)x_2
POL(U21(x1, x2)) = 0
POL(isNatList(x1)) = x_1
POL(zeros) = 0
POL(isNatIList(x1)) = 3 + (4)x_1
POL(U12(x1)) = 1
POL(a__isNat(x1)) = 1
POL(s(x1)) = 0
POL(isNat(x1)) = 0
POL(a__U43(x1)) = 1
POL(a__isNatList(x1)) = x_1
POL(U51(x1, x2, x3)) = (4)x_3
POL(A__LENGTH(x1)) = (2)x_1
POL(a__U61(x1, x2)) = 0
POL(U42(x1, x2)) = x_1 + x_2
POL(a__and(x1, x2)) = x_1 + (2)x_2
POL(U11(x1, x2)) = x_1
POL(0) = 0
POL(a__U12(x1)) = 1
POL(U43(x1)) = 1
POL(a__U51(x1, x2, x3)) = (4)x_3
POL(a__U41(x1, x2, x3)) = 1 + (4)x_3
POL(cons(x1, x2)) = (4)x_2
POL(a__isNatKind(x1)) = 1
POL(U61(x1, x2)) = 0
POL(a__isNatIListKind(x1)) = 1
a__U21(X1, X2) → U21(X1, X2)
a__isNatList(X) → isNatList(X)
a__isNat(X) → isNat(X)
a__U22(X) → U22(X)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U42(X1, X2) → U42(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(tt) → tt
a__zeros → zeros
mark(nil) → nil
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U43(tt) → tt
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U22(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U32(tt) → tt
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNatIList(zeros) → tt
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__U61(tt, L) → s(a__length(mark(L)))
a__U53(tt) → tt
a__isNat(0) → tt
a__isNatList(nil) → tt
a__isNatKind(0) → tt
a__isNatIListKind(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
mark(isNatIList(X)) → a__isNatIList(X)
mark(U43(X)) → a__U43(mark(X))
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U32(X)) → a__U32(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U22(X)) → a__U22(mark(X))
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__and(tt, X) → mark(X)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(length(X)) → a__length(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
A__LENGTH(cons(N, L)) → A__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__AND(tt, X) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
MARK(and(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__U11(tt, V1) → A__ISNATLIST(V1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
A__U21(tt, V1) → A__ISNAT(V1)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(U52(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(and(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatKind(X)) → A__ISNATKIND(X)
MARK(U53(X)) → MARK(X)
A__U31(tt, V) → A__ISNATLIST(V)
MARK(U32(X)) → MARK(X)
MARK(U43(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
MARK(and(X1, X2)) → A__AND(mark(X1), X2)
MARK(U21(X1, X2)) → A__U21(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → A__U52(mark(X1), X2)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(isNat(X)) → A__ISNAT(X)
MARK(isNatList(X)) → A__ISNATLIST(X)
Used ordering: Polynomial interpretation [25,35]:
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__AND(tt, X) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
A__U21(tt, V1) → A__ISNAT(V1)
The value of delta used in the strict ordering is 2.
POL(A__ISNATILISTKIND(x1)) = 0
POL(A__U31(x1, x2)) = 3 + (4)x_2
POL(A__ISNAT(x1)) = 0
POL(a__U11(x1, x2)) = 3 + (3)x_1 + (2)x_2
POL(U41(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3
POL(a__zeros) = 1
POL(mark(x1)) = 0
POL(a__isNatIList(x1)) = 3 + (3)x_1
POL(A__ISNATLIST(x1)) = 0
POL(A__AND(x1, x2)) = (4)x_2
POL(a__length(x1)) = 2 + (2)x_1
POL(tt) = 0
POL(a__U21(x1, x2)) = 2 + (2)x_2
POL(nil) = 1
POL(U31(x1, x2)) = 4 + (3)x_1 + (2)x_2
POL(A__U11(x1, x2)) = 0
POL(isNatKind(x1)) = 2
POL(a__U32(x1)) = 2 + (3)x_1
POL(U22(x1)) = 1 + (4)x_1
POL(a__U31(x1, x2)) = 3 + x_1 + (3)x_2
POL(a__U52(x1, x2)) = 1 + (3)x_1 + (4)x_2
POL(U53(x1)) = 3 + (2)x_1
POL(U32(x1)) = 4 + (4)x_1
POL(length(x1)) = 3 + x_1
POL(a__U22(x1)) = 3 + (3)x_1
POL(U52(x1, x2)) = 1 + (3)x_1 + (4)x_2
POL(a__U53(x1)) = 3 + (2)x_1
POL(A__ISNATKIND(x1)) = 0
POL(a__U42(x1, x2)) = 3 + (2)x_1 + x_2
POL(A__U51(x1, x2, x3)) = 0
POL(and(x1, x2)) = 4 + (3)x_1 + (4)x_2
POL(isNatIListKind(x1)) = 0
POL(U21(x1, x2)) = 4 + (4)x_1 + (2)x_2
POL(isNatList(x1)) = 3 + (3)x_1
POL(zeros) = 0
POL(isNatIList(x1)) = 3 + (3)x_1
POL(U12(x1)) = 1 + (4)x_1
POL(s(x1)) = 1 + x_1
POL(a__isNat(x1)) = 0
POL(A__U52(x1, x2)) = 0
POL(a__U43(x1)) = 3 + (3)x_1
POL(isNat(x1)) = 2 + (4)x_1
POL(U51(x1, x2, x3)) = 4 + (2)x_1 + (3)x_2 + x_3
POL(a__isNatList(x1)) = 3 + (3)x_1
POL(a__U61(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(a__and(x1, x2)) = 3
POL(U42(x1, x2)) = 4 + (3)x_1 + (3)x_2
POL(U11(x1, x2)) = 3 + (4)x_1 + (3)x_2
POL(a__U12(x1)) = 3 + (3)x_1
POL(0) = 0
POL(U43(x1)) = 3 + (2)x_1
POL(a__U51(x1, x2, x3)) = 3 + (2)x_1 + (3)x_2 + (3)x_3
POL(a__U41(x1, x2, x3)) = 2 + (3)x_1 + x_2 + (3)x_3
POL(cons(x1, x2)) = 1 + (2)x_1
POL(A__U21(x1, x2)) = 0
POL(MARK(x1)) = (2)x_1
POL(a__isNatKind(x1)) = 3
POL(U61(x1, x2)) = 2 + (3)x_1 + (3)x_2
POL(a__isNatIListKind(x1)) = 1 + (2)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__AND(tt, X) → MARK(X)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATLIST(cons(V1, V2)) → A__ISNATKIND(V1)
A__ISNAT(s(V1)) → A__ISNATKIND(V1)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__ISNAT(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNATLIST(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__AND(tt, X) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__ISNATKIND(s(V1)) → A__ISNATKIND(V1)
A__ISNATKIND(length(V1)) → A__ISNATILISTKIND(V1)
Used ordering: Polynomial interpretation [25,35]:
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__AND(tt, X) → MARK(X)
The value of delta used in the strict ordering is 4.
POL(A__ISNATILISTKIND(x1)) = 1 + x_1
POL(U41(x1, x2, x3)) = 3 + (3)x_1 + x_2 + (3)x_3
POL(a__zeros) = 3
POL(a__U11(x1, x2)) = 3 + (2)x_2
POL(mark(x1)) = 4
POL(a__isNatIList(x1)) = 3 + (4)x_1
POL(A__AND(x1, x2)) = (4)x_2
POL(a__length(x1)) = 2 + (2)x_1
POL(tt) = 3
POL(a__U21(x1, x2)) = 1 + (4)x_1 + (3)x_2
POL(nil) = 3
POL(U31(x1, x2)) = 3 + (4)x_1 + (3)x_2
POL(isNatKind(x1)) = 3 + (3)x_1
POL(a__U32(x1)) = 3 + (3)x_1
POL(U22(x1)) = 3 + (3)x_1
POL(a__U52(x1, x2)) = 3 + x_1 + (4)x_2
POL(a__U31(x1, x2)) = 1 + (3)x_1 + (4)x_2
POL(U53(x1)) = 3 + (3)x_1
POL(U32(x1)) = 3 + (3)x_1
POL(length(x1)) = 4 + (2)x_1
POL(a__U22(x1)) = 3 + (2)x_1
POL(U52(x1, x2)) = 3 + x_1 + (3)x_2
POL(a__U53(x1)) = 4
POL(A__ISNATKIND(x1)) = 4 + (4)x_1
POL(a__U42(x1, x2)) = 3 + (2)x_2
POL(and(x1, x2)) = 3 + (3)x_1
POL(isNatIListKind(x1)) = 1 + x_1
POL(U21(x1, x2)) = 4 + x_1 + (4)x_2
POL(isNatList(x1)) = 3 + (3)x_1
POL(zeros) = 3
POL(isNatIList(x1)) = 3 + (2)x_1
POL(U12(x1)) = 2 + (3)x_1
POL(a__isNat(x1)) = 3 + (4)x_1
POL(s(x1)) = 1 + x_1
POL(isNat(x1)) = 3 + (3)x_1
POL(a__U43(x1)) = 3 + x_1
POL(a__isNatList(x1)) = 1 + (3)x_1
POL(U51(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (2)x_3
POL(a__U61(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(a__and(x1, x2)) = 3 + x_1 + (4)x_2
POL(U42(x1, x2)) = 1 + (3)x_1 + x_2
POL(U11(x1, x2)) = 2 + (2)x_1
POL(0) = 3
POL(a__U12(x1)) = 2
POL(U43(x1)) = 3 + x_1
POL(a__U51(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3
POL(a__U41(x1, x2, x3)) = 2 + x_1 + (2)x_2 + (3)x_3
POL(cons(x1, x2)) = 3 + (4)x_1 + (4)x_2
POL(MARK(x1)) = x_1
POL(a__isNatKind(x1)) = 3 + (3)x_1
POL(U61(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(a__isNatIListKind(x1)) = 3 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
A__ISNATILISTKIND(cons(V1, V2)) → A__ISNATKIND(V1)
A__AND(tt, X) → MARK(X)
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__AND(tt, X) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(isNatIListKind(X)) → A__ISNATILISTKIND(X)
A__AND(tt, X) → MARK(X)
Used ordering: Polynomial interpretation [25,35]:
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
The value of delta used in the strict ordering is 4.
POL(A__ISNATILISTKIND(x1)) = 4 + (4)x_1
POL(U41(x1, x2, x3)) = 3 + (4)x_1 + x_2 + (3)x_3
POL(a__zeros) = 1
POL(a__U11(x1, x2)) = 3 + (3)x_1 + (2)x_2
POL(mark(x1)) = 2 + (3)x_1
POL(a__isNatIList(x1)) = 3
POL(A__AND(x1, x2)) = 4 + (3)x_2
POL(a__length(x1)) = 2 + (3)x_1
POL(tt) = 0
POL(a__U21(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(nil) = 3
POL(U31(x1, x2)) = 3 + x_1 + (2)x_2
POL(isNatKind(x1)) = 3 + (3)x_1
POL(a__U32(x1)) = 3
POL(U22(x1)) = 1 + (4)x_1
POL(a__U52(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(a__U31(x1, x2)) = 4 + (3)x_1 + x_2
POL(U53(x1)) = 3 + (3)x_1
POL(U32(x1)) = 2 + (3)x_1
POL(length(x1)) = 3 + (3)x_1
POL(a__U22(x1)) = 3 + (3)x_1
POL(U52(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(a__U53(x1)) = 3 + (3)x_1
POL(a__U42(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(and(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(isNatIListKind(x1)) = 4 + (3)x_1
POL(U21(x1, x2)) = 3 + (2)x_1 + x_2
POL(isNatList(x1)) = 1 + (3)x_1
POL(zeros) = 3
POL(isNatIList(x1)) = 3 + (3)x_1
POL(U12(x1)) = 2 + (3)x_1
POL(a__isNat(x1)) = 3 + (3)x_1
POL(s(x1)) = 4 + (3)x_1
POL(isNat(x1)) = 3 + (3)x_1
POL(a__U43(x1)) = 2 + (2)x_1
POL(a__isNatList(x1)) = 3 + (4)x_1
POL(U51(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3
POL(a__U61(x1, x2)) = 2 + (3)x_1 + (3)x_2
POL(a__and(x1, x2)) = 2 + x_1 + (3)x_2
POL(U42(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(U11(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(0) = 3
POL(a__U12(x1)) = 2
POL(U43(x1)) = 3 + (3)x_1
POL(a__U51(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3
POL(a__U41(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + x_3
POL(cons(x1, x2)) = 3 + x_1 + (4)x_2
POL(MARK(x1)) = (2)x_1
POL(a__isNatKind(x1)) = 3
POL(U61(x1, x2)) = 3 + (3)x_1 + (4)x_2
POL(a__isNatIListKind(x1)) = 2 + x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
A__ISNATILISTKIND(cons(V1, V2)) → A__AND(a__isNatKind(V1), isNatIListKind(V2))
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
A__U51(tt, V1, V2) → A__ISNAT(V1)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__U51(tt, V1, V2) → A__ISNAT(V1)
A__U51(tt, V1, V2) → A__U52(a__isNat(V1), V2)
A__U11(tt, V1) → A__ISNATLIST(V1)
A__U52(tt, V2) → A__ISNATLIST(V2)
Used ordering: Polynomial interpretation [25,35]:
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
The value of delta used in the strict ordering is 1.
POL(A__ISNAT(x1)) = 1 + (2)x_1
POL(U41(x1, x2, x3)) = 3 + (2)x_1 + (2)x_2 + x_3
POL(a__zeros) = 2
POL(a__U11(x1, x2)) = 1 + x_1 + (3)x_2
POL(mark(x1)) = 1
POL(a__isNatIList(x1)) = 3 + (3)x_1
POL(A__ISNATLIST(x1)) = x_1
POL(a__length(x1)) = 3 + (3)x_1
POL(tt) = 0
POL(a__U21(x1, x2)) = 1 + (2)x_2
POL(nil) = 3
POL(A__U11(x1, x2)) = 1 + x_2
POL(U31(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(isNatKind(x1)) = 3 + (3)x_1
POL(a__U32(x1)) = 3 + (3)x_1
POL(U22(x1)) = 3 + (3)x_1
POL(a__U52(x1, x2)) = 3 + (3)x_1 + (4)x_2
POL(a__U31(x1, x2)) = 3 + (2)x_1 + (3)x_2
POL(U53(x1)) = 4 + (3)x_1
POL(U32(x1)) = 3 + (4)x_1
POL(length(x1)) = (4)x_1
POL(a__U22(x1)) = 2 + (3)x_1
POL(U52(x1, x2)) = 4
POL(a__U53(x1)) = 1 + x_1
POL(a__U42(x1, x2)) = 3 + (2)x_1 + (4)x_2
POL(A__U51(x1, x2, x3)) = 2 + (2)x_2 + x_3
POL(and(x1, x2)) = 3 + x_1 + (3)x_2
POL(U21(x1, x2)) = 2 + x_2
POL(isNatIListKind(x1)) = 1
POL(isNatList(x1)) = 1 + (3)x_1
POL(zeros) = 2
POL(isNatIList(x1)) = 4 + (3)x_1
POL(U12(x1)) = 3 + (2)x_1
POL(a__isNat(x1)) = 3 + (2)x_1
POL(s(x1)) = x_1
POL(A__U52(x1, x2)) = 1 + x_2
POL(isNat(x1)) = 3 + (2)x_1
POL(a__U43(x1)) = 4 + x_1
POL(a__isNatList(x1)) = 2 + x_1
POL(U51(x1, x2, x3)) = 2 + x_1 + (4)x_2 + (2)x_3
POL(a__U61(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(a__and(x1, x2)) = 3
POL(U42(x1, x2)) = 4 + x_1 + (2)x_2
POL(U11(x1, x2)) = 3 + x_1 + (2)x_2
POL(0) = 3
POL(a__U12(x1)) = 3 + (4)x_1
POL(U43(x1)) = 2 + (3)x_1
POL(a__U51(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3
POL(a__U41(x1, x2, x3)) = 3 + (3)x_1 + x_2 + (3)x_3
POL(cons(x1, x2)) = 2 + (4)x_1 + (2)x_2
POL(A__U21(x1, x2)) = 1 + (2)x_2
POL(a__isNatKind(x1)) = 1
POL(U61(x1, x2)) = 3 + (2)x_1 + (3)x_2
POL(a__isNatIListKind(x1)) = 4
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
A__ISNATLIST(cons(V1, V2)) → A__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
A__ISNAT(length(V1)) → A__U11(a__isNatIListKind(V1), V1)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
A__ISNAT(s(V1)) → A__U21(a__isNatKind(V1), V1)
A__U21(tt, V1) → A__ISNAT(V1)
The value of delta used in the strict ordering is 3.
POL(A__ISNAT(x1)) = 1 + (4)x_1
POL(a__zeros) = 3
POL(U41(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (2)x_3
POL(a__U11(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(mark(x1)) = 1 + (4)x_1
POL(a__isNatIList(x1)) = 3 + x_1
POL(a__length(x1)) = 4 + x_1
POL(tt) = 1
POL(a__U21(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(nil) = 3
POL(U31(x1, x2)) = 4 + (4)x_1 + (2)x_2
POL(isNatKind(x1)) = 3
POL(a__U32(x1)) = 4 + (3)x_1
POL(U22(x1)) = 2 + (2)x_1
POL(a__U52(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(a__U31(x1, x2)) = 1 + (3)x_1 + (3)x_2
POL(U53(x1)) = 3 + (3)x_1
POL(U32(x1)) = 3 + (3)x_1
POL(length(x1)) = (3)x_1
POL(a__U22(x1)) = 3 + (3)x_1
POL(U52(x1, x2)) = 2 + (3)x_1 + (3)x_2
POL(a__U53(x1)) = 3 + (3)x_1
POL(a__U42(x1, x2)) = 3 + (4)x_1 + (3)x_2
POL(and(x1, x2)) = 3 + x_1 + x_2
POL(U21(x1, x2)) = 3 + x_1 + (3)x_2
POL(isNatIListKind(x1)) = 3 + (2)x_1
POL(isNatList(x1)) = 2 + (3)x_1
POL(zeros) = 3
POL(isNatIList(x1)) = 3 + (3)x_1
POL(U12(x1)) = 3 + x_1
POL(s(x1)) = 2 + (4)x_1
POL(a__isNat(x1)) = 1 + (3)x_1
POL(isNat(x1)) = 1 + x_1
POL(a__U43(x1)) = 3 + (3)x_1
POL(a__isNatList(x1)) = 2 + (3)x_1
POL(U51(x1, x2, x3)) = 3 + (2)x_1 + (3)x_2 + (3)x_3
POL(a__U61(x1, x2)) = 1 + (3)x_1 + (2)x_2
POL(U42(x1, x2)) = 3 + (3)x_1 + (2)x_2
POL(a__and(x1, x2)) = 2 + (4)x_2
POL(U11(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(0) = 3
POL(a__U12(x1)) = 3 + (3)x_1
POL(U43(x1)) = 3 + (3)x_1
POL(a__U51(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (3)x_3
POL(cons(x1, x2)) = (4)x_1
POL(a__U41(x1, x2, x3)) = 3 + (3)x_1 + (3)x_2 + (2)x_3
POL(A__U21(x1, x2)) = 4 + (4)x_2
POL(a__isNatKind(x1)) = 3
POL(U61(x1, x2)) = 3 + (3)x_1 + (3)x_2
POL(a__isNatIListKind(x1)) = 3 + (3)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
a__zeros → cons(0, zeros)
a__U11(tt, V1) → a__U12(a__isNatList(V1))
a__U12(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, V) → a__U32(a__isNatList(V))
a__U32(tt) → tt
a__U41(tt, V1, V2) → a__U42(a__isNat(V1), V2)
a__U42(tt, V2) → a__U43(a__isNatIList(V2))
a__U43(tt) → tt
a__U51(tt, V1, V2) → a__U52(a__isNat(V1), V2)
a__U52(tt, V2) → a__U53(a__isNatList(V2))
a__U53(tt) → tt
a__U61(tt, L) → s(a__length(mark(L)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(length(V1)) → a__U11(a__isNatIListKind(V1), V1)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatIList(V) → a__U31(a__isNatIListKind(V), V)
a__isNatIList(zeros) → tt
a__isNatIList(cons(V1, V2)) → a__U41(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__isNatIListKind(nil) → tt
a__isNatIListKind(zeros) → tt
a__isNatIListKind(cons(V1, V2)) → a__and(a__isNatKind(V1), isNatIListKind(V2))
a__isNatKind(0) → tt
a__isNatKind(length(V1)) → a__isNatIListKind(V1)
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__isNatList(nil) → tt
a__isNatList(cons(V1, V2)) → a__U51(a__and(a__isNatKind(V1), isNatIListKind(V2)), V1, V2)
a__length(nil) → 0
a__length(cons(N, L)) → a__U61(a__and(a__and(a__isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNatList(X)) → a__isNatList(X)
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2)) → a__U42(mark(X1), X2)
mark(U43(X)) → a__U43(mark(X))
mark(isNatIList(X)) → a__isNatIList(X)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2)) → a__U52(mark(X1), X2)
mark(U53(X)) → a__U53(mark(X))
mark(U61(X1, X2)) → a__U61(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatIListKind(X)) → a__isNatIListKind(X)
mark(isNatKind(X)) → a__isNatKind(X)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNatList(X) → isNatList(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__isNat(X) → isNat(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2) → U42(X1, X2)
a__U43(X) → U43(X)
a__isNatIList(X) → isNatIList(X)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2) → U52(X1, X2)
a__U53(X) → U53(X)
a__U61(X1, X2) → U61(X1, X2)
a__length(X) → length(X)
a__and(X1, X2) → and(X1, X2)
a__isNatIListKind(X) → isNatIListKind(X)
a__isNatKind(X) → isNatKind(X)